A Successive Minima Method for Implicit Approximation

This paper is concerned with the problem of approximating a collection of unorganized data points by an algebraic tensor-product B-spline curve. The implicit approximation to data points would be ideally based on minimizing the sum of squares of geometric distance. Since the geometric distance from a point to an implicit curve cannot be computed analytically, Sampson distance, which is the first-order approximation of the geometric distance, is introduced via a derivation from the viewpoint of optimization theory. Then, the implicit approximation is modeled as a nonlinear optimization problem by minimizing the Sampson error and the fair term for smoothing effect. By the idea of successive minima technique, we induct a quadratic constraint function of the data at every iteration step, and show that the minimization reduces to a constrained quadratic optimization subproblem, which can be solved as generalized eigenvector fitting. This successive procedure is stable and computationally reasonable. Some examples are implemented in our approach, and the high-quality reconstruction curve is obtained in a robust way.